Heegner Zeros of Theta Functions ( Trans . Ams . 355 ( 2003 ) , No

Heegner divisors play an important role in number theory. However little is known on whether a modular form has Heegner zeros. In this paper, we start to study this question for a family of classical theta functions, and prove a quantitative result, which roughly says that many of these theta functions have a Heegner zero of discriminant −7. This leads to some interesting questions on the arithmetic of certain elliptic curves, which we also address here. 0. Introduction. Let N ≥ 1 be an integer and let f be a non-zero meromorphic modular form of level N with algebraic Fourier coefficients. Then f can be viewed as a (meromorphic) section of a line bundle on the modular curve X0(N) and thus its zeros and poles give a divisor in X0(N) which is algebraic. These important divisors appear in the beautiful works of Rohrlich ([R]) on Jensen’s formula and more recently of Bruinier, Kohnen, and Ono ([B-K-O]) on the values of modular functions. However, if we let τ be a zero or a pole of f on the upper half plane H, then it is well-known that τ is either quadratic (Heegner point) or transcendental. So it is very interesting to isolate and understand the Heegner zeros/poles of f . We recall that a Heegner point on X0(N) of discriminant −D is represented by a quadratic number τ = b+ √−D 2aN with integers a > 0 and b. Although Heegner points play very important roles in many branches of number theory, such as the Gross-Zagier formula, Kolyvagin’s Euler system, and the Borcherds product theory, to name a few, little is known about the Heegner zeros of modular forms. 1991 Mathematics Subject Classification. 11G05 11M20 14H52.